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Statistics

1.Making a frequency table.
A frequency table is a table showing a record of raw data arranged in preparation for analysis and interpretation to give meaningful information.
It's constructed by arranging collected data in ascending order of magnitude with their corresponding frequencies.

1.Making a frequency table.A frequency table is a table showing a record of
raw data arranged in preparation for analysis and interpretation to give
meaningful information.
It's constructed by arranging collected data in ascending order of
magnitude with their corresponding frequencies.

Example 1The following data shows the number of goals
scored by a certain team during a football premiership league matches
0,0,1,1,2,1,1,0,3,4,4,3,2,2,5,6,4,6,4,1,1,1,4,5,5.
Make a frequency table showing the distribution above.

To draw a frequency table.

Step 1:

Construct a table with three columns. The first column to read the
number of goals(x). write down all the data values in ascending order of
magnitude i.e.(0-6)

Step 2:

The second column to read tally, then place a tally mark at the
appropriate place in the column for every data value.
When the fifth tally is reached, draw a line across the first four as
shown
,

Continue this process until all data values in the list are tallied.

Step 3:

Count the number of tally marks for each data value and write it in
the third column as frequency.

Solution

Grouped Data

Example 2The following data shows the distribution of
marks out of 50 in a mathematics test for 20 students.
2,7,12,11,13,14,14,4,18,19,20,15,12,23,22,32,34,31,38,37.
Makes a grouped frequency table for the above data using a class
interval of 5.

Solution

Measures of central tendency: mean, median
and mode.The mean, Mode and median are values about which
the distribution of a set of data is considered to be roughly balanced.MeanThis is the sum of data values divided by the
number of data values. The mean can be obtained for grouped or ungrouped
data.

ExampleThe following are ages of college students.
Calculate their mean age 27, 28, 31, 30, 32, 35, 28, 30, 29

Solution

Grouped data

ExampleThe table below shows the distribution of
average marks out of 50 for thirty physics students in ngombini school.

Calculate the mean mark for the thirty
students.

SolutionFirst make a frequency table as shown below.

MedianThe median of a set of data values is the middle
values of the set of data when it has been arranged in ascending order.

SolutionArrange the data values in ascending order
22, 24, 25, 25, 26, 27, 28, 29, 37
Since the data values are 9, the fifth value is the middle value.Therefore the median of the set of data is 26.
The number of the values n, in the data set = 9.

ModeThe mode is the most frequent occurring value in
the data.

Example

Find the mode of the following set of data
38, 34, 38, 35, 32, 38, 39.

Solution
The mode is 38 since it occurs most often.

NoteIt's possible for a set of data to have more
than one mode.

STATISTICS

Statistics is a very vital branch of
mathematics as it involves gathering of raw data, recording, analysis
and interpretation of data, into meaningfull information, which helps us
to make proper judgment in life.

The Assumed Mean

When the number of values in a set of data is large, the assumed mean
method is used. The assumed mean is an expected value of the mean also
known as the working mean, usually denoted as A. The assumed mean does
not need to be one of the values given.

Estimate by calculation:1.The median
2.The lower and upper quartilesSolution
For one to estimate the median and the two quartiles, one must create a
cumulative frequency table with lower and upper class boundary column as
shown.

The Median

For us to calculate the median we must first
of all identify the median class, and since we have 100 data values the
middle value must lie at the 1/2 (N +1)value (refer to background) i.e.
it must lie at the 1/2(100+150).5th
value in the cumulative frequency column.
Therefore the median class should be the one having 59.5-69.5 class
boundaries

The lower quartile (Q1)Identify the quartile class as you did with the
median class above.Calculate the lower quartile (Q1)
using the formula

Where
L= Lower class boundary of the Q1 class.
N= Total frequency of the distribution.
Cfa = Cumulative frequency above the Q1
class.
fQ1= Frequency of the Q1
class.
i= Class interval of the Q1 class.

In the case above:

L=39.5
N=100
Cfa= 24
fQ1 = 12
i = 10

For the Upper Quartile (Q3)Identify the upper quartile class (Q3)
and highlight it.Calculate the upper quartile Q3
using the formula

where
L= Lower class boundary of the Q3 class.
N= Total frequency of the distribution.
Cfa = Cumulative frequency above the Q3
class.
fQ3= Frequency of the Q3
class.
i= Class interval of the Q3 class.

In the case above:

L=69.5
N=100
Cfa= 68
fQ3 = 15
i = 10

NB:The difference between the upper
quartile (Q3) and the lower quartile (Q1)is
called the interquartile range.
Therefore, in the case above the interquartile range is given by

Interquartile range =Q3-
Q1
= 74.17 - 40.33
= 33.84

Half of the interquartile range is also known as the
semi-interquartile range or the quartile deviation, i.e.
semi-interquartile range (quartile deviation)

Use of Ogive/ Cumulative Frequency Curve.An ogive/ cumulative frequency curve is obtained
by plotting cumulative frequency values against the upper class
boundaries.
The curve can be used to estimate the median, the lower quartile and the
upper quartile.ExampleThe table below represents mass of sugar in kg
consumed in a college per day.

Example 1One hundred students were asked to estimate the
distance between their dormitories and their classes in meters. Their
results were shown as in the table below.

SolutionUsing steps 1 to 5 in II above and taking 38 to
be the assumed mean, the deviation ,d is given by x-A=d.

Example 2

The height of 50 tree seedlings in a tree nursery in cm were recorded
as in the table below. Using a suitable assumed mean estimate the mean
to the nearest cm.

SolutionPrepare a frequency table and obtain the
midpoint(x) of each class. Follow steps 1 to 5 as previously explained.(For grouped data we get the midpoint of each
class first. The deviation are then calculated by subtracting the
assumed mean A=28 from each midpoint x. ie 3-28= -25, 8-28=-20, ---,
48-28= +20)

NB: the assumed mean in both examples must be permanently highlighted.

Making a cumulative frequency tableCumulative frequency table is made by adding a
cumulative frequency column to a frequency table.
Cumulative frequency is the gradual increase of frequencies as a result
of subsequent addition.

Example.The table below shows the distribution of masses
in kg of 35 people. Make a cummulative frequency table.

Solution

Estimating the median and the quartilesa) CalculationMedian is the value that divides a distribution
of data values into two equal parts.
Quartiles (Q) are the values that divide a distribution into four equal
parts. Thus the lower quartile(Q1)
is the value below which lies the 25% (1/4) of the distribution and the
upper quartile(Q3)
is the value below which lies the 75% (3/4) of the distribution.

ExampleThe table below shows the speed of 100 public
service vehicles during a police check.

.

Statistics

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